Is the first Hardy-Littlewood conjecture true?
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The first Hardy-Littlewood conjecture, also called the k-tuple conjecture, gives an asymptotic formula for the number of primes p ≤ x such that p, p + a₁, p + a₂, ..., p + aₖ are all prime, for any tuple (a₁, a₂, ..., aₖ).

For the exact formula, see: https://mathworld.wolfram.com/k-TupleConjecture.html

k-Tuple Conjecture -- from Wolfram MathWorld
The first of the Hardy-Littlewood conjectures. The k-tuple conjecture states that the asymptotic number of prime constellations can be computed explicitly. In particular, unless there is a trivial divisibility condition that stops p, p+a_1, ..., p+a_k from consisting of primes infinitely often, then such prime constellations will occur with an asymptotic density which is computable in terms of a_1, ..., a_k. Let 0<m_1<m_2<...<m_k, then the k-tuple conjecture predicts that the...
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